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BAGHDAD — When al-Qaida-style insurgents overran the northern city of Mosul, part of the war booty they seized were what they claimed were five U.S.-made helicopters.

Noting the copters were nearly new, the group said in a posting on its Twitter feed: “We’ll expect the Americans to honor the warranty and service them for us.”

“Not only are they effective jihadists, but they have a sense of humor,” said Toby Dodge, director of the Middle East Centre at the London School of Economics, who related that anecdote.

Behind the image of savagery the extremists of the Islamic State of Iraq and Syria present to the world, as casual executioners who kill helpless prisoners and behead rival jihadis, lies a disciplined organization that employs social media and sophisticated financial strategies in the funding and governance of the areas it has conquered.

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The insurgents seized up to $400 million from the central bank in Mosul, said Atheel Nujaifi, governor of Nineveh province, and reportedly emptied the vaults in all the other banks in a city of more than 1 million residents. Other officials cite lower figures when discussing the central-bank theft.

In a bloody seesaw battle for control of Iraq’s biggest oil refinery at Baiji, halfway between Baghdad and Mosul, the insurgents worked with the families of employees there to broker a cease-fire so the workers could be safely evacuated.

It was no humanitarian gesture. “They want them to run the refinery when the fighting is over,” said one local official, who spoke on condition of anonymity out of fear the Islamic State of Iraq and Syria (ISIS), also known as the Islamic State of Iraq and the Levant, would kill him.

Their extortion rackets in Mosul netted up to $8 million a month, Gen. Mahdi Gharawi, until recently the Nineveh province police commander, said in an interview with Niqash, an Arabic language news website. And that was before they took over. Once in charge, they typically levy “taxes,” which are just as lucrative. Road taxes of $200 on trucks are collected all over northern Iraq to allow the trucks safe passage. The Iraqi government claims they are levying a “tax” on Christians in Mosul, who were a significant minority there, to avoid being crucified.

Even a popular cellphone app that helped the Islamic State propel its Twitter feed to the top of the jihadi charts had advertising embedded in it. All of that, in turn, was part of a savvy social-media campaign to persuade well-heeled Saudi and Persian Gulf Arab supporters to donate to their operations. “We don’t know the exact amount of money they stole from Mosul,” said Ameen Hadi, a member of the last Iraqi Parliament’s finance committee. “But it is big, big enough that ISIS can use it to occupy other countries too.”

A member of the board of governors of the Central Bank of Iraq was reluctant to say how much the Islamic State got away with in Mosul, but said it was at least $85 million and could have been much more. He spoke on the condition of anonymity.

“ISIS gets some money from outside donors, but that pales in comparison to their self-funding,” said a U.S. counterterrorism official.

While the Islamic State “is among the wealthiest terrorist groups on the planet,” the official said, “it also has significant expenses. Resources flowing into the group’s coffers tend to move out the door in the form of payments fairly quickly.”

The militant group has so much cash that it has reopened some of the banks it earlier looted in Fallujah to stash it in. Jassim Ahmed, 35, who works as a taxi driver in the city, which has been under militant control since January, said he asked one of the gunmen guarding the banks where the militants get their money. “Don’t ask me again,” the gunman told him, he said. “Just understand, we have a budget to administer all of Iraq, not just Anbar.”

“We no longer have to imagine a terror state,” said Kamel Wazne, a Beirut-based analyst who has followed the group’s development into a self-financing, territory-controlling entity. “We have one.”

The Islamic State started amassing a bankroll in Syria last year after it took over the eastern Syrian oil fields, near Raqqa. It operates primitive refineries to produce products for local use by Islamic State fighters, but sells much of the crude to its enemy: the Syrian government. In Minbij, it runs a local cement factory, and in Raqqa, merchants even pay the militants a trash-collection fee.

Invading Iraq has just expanded the revenue base. “The more territory they hold, the more they will become self-reliant,” said Peter Neumann, a professor of security studies at King’s College London. “That is one of the dangers and why they have to be stopped. If they become self-reliant and can start paying people salaries and such, that makes it much harder to dislodge them.”

The organization is always on the lookout for new revenue streams. The group’s extreme ideology may call for all non-Sunni Muslims to be killed, but this is apparently overlooked when there is money to be made on the backs of unbelievers. In the past week, several groups of Turkish workers, a group of 40 Indian workers and a Chinese official were among foreigners who disappeared in territory that the Islamic State overran and were later released unharmed. While no one confirmed that ransoms were paid, kidnapping for ransom has been part of the militants’ business plan.

So far the extremists have been prevented from expanding their operations into Iraq’s oil-producing areas. But for the past week they have besieged the Baiji oil refinery, the largest in the country with a capacity of 310,000 barrels a day, with the facility briefly falling completely under their control Wednesday.

While many experts question whether the insurgents could operate the sophisticated facility, they could bring its Iraqi workers back or, at least, gain access to numerous storage tanks.

Baiji supplies Iraq with a third of its domestic fuels, and a nearby 600-megawatt power plant provides 10 percent of the country’s electricity, according to Barclay’s Research. All of the provinces where the Islamic State has been active have been without electricity since Baiji’s shutdown.

Fear of gasoline shortages sent many Iraqis to Kurdistan this week, where long lines formed Friday at gas stations, which were ordered to give no one more than about 8 gallons apiece.

“If Baiji falls, the fuel crisis will be huge,” said Barham Salih, the former prime minister of the Kurdish regional government.

Iraqi government employees, even in areas taken over by the militants, continue to draw their salaries, and the government said they would be allowed to go to safe areas to collect their pay.

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